3D dynamics

The characteristic polynomial of a 3 x 3 matrix A can written as

$${\rm det} \left[ A - \lambda I_{3}\right] = -\lambda^3 + \lambda^2 {\rm tr}(A) + \lambda \frac{1}{2}\left[ {\rm tr}(A^2) - {\rm tr}^2(A) \right] + {\rm det}(A) $$.

We note that the condition for a trinomial of the form

$$ \lambda^3 + a_1 \lambda^2 + a_2 \lambda + a_3 = 0 \, $$

to have three roots with negative real parts we need

$$ a_1 > 0, a_2 > 0, a_1 a_2 > a_3 \, $$

By reversing these three inequalities we arrive at the condition for three positive real parts; $$ a_1 > 0, a_2 > 0, a_1 a_2 > a_3 \, $$

Three negative roots of course corresponds to a sink while three positive roots corresponds to a source. If neither of the above conditions are satisfied, we are in the situation of a saddle point.

There are methods to determine whether there are two roots with non-zero imaginary components, but we do not present them here.